arXiv subjects

Dinghao Zhu

Publications and source records attributed to Dinghao Zhu.

3 recordsLinked to original sources

arXiv2026

The direct scattering problem for the defocusing nonlinear Schr\"odinger equation with step-like periodic background

We consider the defocusing nonlinear Schr\"odinger equation for step-like data connecting two, in general different, genus-one periodic states. Starting from the Jost solutions associated with the left and right backgrounds, we determine the corresponding scattering data and reorganize them through a scalar conjugation and a suitable symmetrization. The resulting formulation identifies the inverse problem with a full dark-soliton gas Riemann--Hilbert problem supplemented by a radiative jump on the real axis. In this way, the effects of the two periodic backgrounds are separated, and a precise link is established between step-like periodic scattering and the full-gas construction. Assuming the absence of discrete eigenvalues and appropriate regularity of the scattering data, we establish the existence and uniqueness of the associated Riemann--Hilbert problem.

math.AP↗

arXiv2026

Arbitrary-genus dark soliton gases in the defocusing nonlinear Schr\"{o}dinger hydrodynamics

The defocusing nonlinear Schr\"{o}dinger hydrodynamics supports exact dark solitons under finite density boundary conditions. However, the dark soliton gas, an interacting ensemble of dark solitons, has not yet been studied. In this work, we introduce an arbitrary-genus potential of dark soliton gases by considering the limit of the $\mathcal{N}$-dark soliton as $\mathcal{N}\to \infty$. The large-space asymptotics and long-time evolution of this dark soliton gas potential are analytically investigated through Deift-Zhou nonlinear steepest descent approach. The genus-$N$ dark soliton gas potential approaches the genus-$N$ finite-gap solution as $x \to -\infty$ and the background $1$ as $x \to +\infty$. In the long-time evolution, as the self-similar variable $\xi=x/t$ increases, the gas configuration exhibits a cascade of behaviours, passing from unmodulated and modulated genus-$N$ regions and progressively reducing the genus down to the planar region (unmodulated genus-$0$ region). Notably, the evolution of lower-genus soliton gases can be embedded within that of higher-genus gases, exhibiting identical dynamics within specific regimes. This phenomenon is encoded by the underlying spectra. We also include numerical validations, in perfect agreement with the theoretical predictions.

math-ph↗

arXiv2024

Genus two KdV soliton gases and their long-time asymptotics

This paper employs the Riemann-Hilbert problem to provide a comprehensive analysis of the asymptotic behavior of the high-genus Korteweg-de Vries soliton gases. It is demonstrated that the two-genus soliton gas is related to the two-phase Riemann-Theta function as $x \to +\infty$, and approaches to zero as $x \to -\infty$. Additionally, the long-time asymptotic behavior of this two-genus soliton gas can be categorized into five distinct regions in the $x$-$t$ plane, which from left to right are rapidly decay, modulated one-phase wave, unmodulated one-phase wave, modulated two-phase wave, and unmodulated two-phase wave. Moreover, an innovative method is introduced to solve the model problem associated with the high-genus Riemann surface, leading to the determination of the leading terms, which is also related with the multi-phase Riemann-Theta function. A general discussion on the case of arbitrary $N$-genus soliton gas is also presented.

nlin.SI↗